7,760 research outputs found

    Preimage problems for deterministic finite automata

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    Given a subset of states SS of a deterministic finite automaton and a word ww, the preimage is the subset of all states mapped to a state in SS by the action of ww. We study three natural problems concerning words giving certain preimages. The first problem is whether, for a given subset, there exists a word \emph{extending} the subset (giving a larger preimage). The second problem is whether there exists a \emph{totally extending} word (giving the whole set of states as a preimage)---equivalently, whether there exists an \emph{avoiding} word for the complementary subset. The third problem is whether there exists a \emph{resizing} word. We also consider variants where the length of the word is upper bounded, where the size of the given subset is restricted, and where the automaton is strongly connected, synchronizing, or binary. We conclude with a summary of the complexities in all combinations of the cases

    Reduction of dynamical biochemical reaction networks in computational biology

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    Biochemical networks are used in computational biology, to model the static and dynamical details of systems involved in cell signaling, metabolism, and regulation of gene expression. Parametric and structural uncertainty, as well as combinatorial explosion are strong obstacles against analyzing the dynamics of large models of this type. Multi-scaleness is another property of these networks, that can be used to get past some of these obstacles. Networks with many well separated time scales, can be reduced to simpler networks, in a way that depends only on the orders of magnitude and not on the exact values of the kinetic parameters. The main idea used for such robust simplifications of networks is the concept of dominance among model elements, allowing hierarchical organization of these elements according to their effects on the network dynamics. This concept finds a natural formulation in tropical geometry. We revisit, in the light of these new ideas, the main approaches to model reduction of reaction networks, such as quasi-steady state and quasi-equilibrium approximations, and provide practical recipes for model reduction of linear and nonlinear networks. We also discuss the application of model reduction to backward pruning machine learning techniques

    A generalization of a theorem of Hurewicz for quasi-Polish spaces

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    We identify four countable topological spaces S2S_2, S1S_1, SDS_D, and S0S_0 which serve as canonical examples of topological spaces which fail to be quasi-Polish. These four spaces respectively correspond to the T2T_2, T1T_1, TDT_D, and T0T_0-separation axioms. S2S_2 is the space of rationals, S1S_1 is the natural numbers with the cofinite topology, SDS_D is an infinite chain without a top element, and S0S_0 is the set of finite sequences of natural numbers with the lower topology induced by the prefix ordering. Our main result is a generalization of Hurewicz's theorem showing that a co-analytic subset of a quasi-Polish space is either quasi-Polish or else contains a countable Π20\Pi^0_2-subset homeomorphic to one of these four spaces

    Ordered Reference Dependent Choice

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    This paper studies how violations of structural assumptions like expected utility and exponential discounting can be connected to rationality violations that arise from reference-dependent preferences, even if behavior is fully standard when the reference is fixed. A reference-dependent generalization of arbitrarily behavioral postulates captures changing preferences across choice domains. It gives rise to a linear order that endogenously determines reference alternatives, which in turn determines the preference parameters for a choice problem. With canonical models as backbones, preference changes are captured using known technologies like the concavity of utility functions and the levels of discount factors. The framework allows us to study risk, time, and social preferences collectively, where seemingly independent anomalies are interconnected through the lens of reference-dependent choice

    Improving the Upper Bound on the Length of the Shortest Reset Word

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    We improve the best known upper bound on the length of the shortest reset words of synchronizing automata. The new bound is slightly better than 114 n^3 / 685 + O(n^2). The Cerny conjecture states that (n-1)^2 is an upper bound. So far, the best general upper bound was (n^3-n)/6-1 obtained by J.-E. Pin and P. Frankl in 1982. Despite a number of efforts, it remained unchanged for about 35 years. To obtain the new upper bound we utilize avoiding words. A word is avoiding for a state q if after reading the word the automaton cannot be in q. We obtain upper bounds on the length of the shortest avoiding words, and using the approach of Trahtman from 2011 combined with the well-known Frankl theorem from 1982, we improve the general upper bound on the length of the shortest reset words. For all the bounds, there exist polynomial algorithms finding a word of length not exceeding the bound
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